A101200 Number of partitions of n with rank 3 (the rank of a partition is the largest part minus the number of parts).

0, 0, 0, 1, 0, 1, 1, 2, 2, 4, 3, 6, 7, 10, 11, 17, 18, 26, 30, 40, 47, 63, 72, 94, 111, 140, 165, 209, 244, 304, 359, 440, 519, 634, 743, 901, 1060, 1273, 1494, 1789, 2092, 2491, 2914, 3449, 4026, 4752, 5530, 6502, 7561, 8852, 10272, 11997, 13889, 16171, 18695, 21700, 25041, 29002

Offset: 1

Author

Emeric Deutsch, Dec 12 2004

Keywords

• nonn

Comments

Column k=3 in the triangle A063995.

Examples

			a(6)=1 because the 11 partitions 6,51,42,411,33,321,3111,222,2211,21111,111111 have ranks 5,3,2,1,1,0,-1,-1,-2,-3,-5, respectively.
		

References

• George E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000041, A063995, A101198, A101199.

Programs

Maple

with(combinat): for n from 1 to 45 do P:=partition(n): c:=0: for j from 1 to nops(P) do if P[j][nops(P[j])]-nops(P[j])=3 then c:=c+1 else c:=c fi od: a[n]:=c: od: seq(a[n],n=1..45);

Mathematica

Table[Count[IntegerPartitions[n],?(#[[1]]-Length[#]==3&)],{n,60}] (* _Harvey P. Dale, Feb 11 2025 *)

Extensions

More terms, Joerg Arndt, Oct 07 2012